Need help with these calculus questions

Question

1)on what intervals is this derivative negative? fprime(x)=3x(x-2)(x-6). Justify your answer

2)On what intervals is f prime(x) greater than O if f(x)2x+8/x. Justify your answer.

3) Find the rate of change of the volume of a ball with respect to time when the radius is 10 cm if the radius changes at a constant rate of 2cm/sec (v= 4/3πr^3)

Answer 1

For 3)

 

We want to find   dV/dt

 

Given:

r = 10

dr/dt = 2

 

Therefore,

 

dV/dt = (dV/dr) * (dr/dt)

 

 

First, find dV/dr.

 

dV/dr = 4πr² = 4π(10)² = 400π

 

dV/dt = 400π * 2

dV/dt  = 800π cm³ / s

Answer 2

For 1), we can see that the derivative is a product of 3 functions, 3x (which is positive for x>0), (x-2) which is positive for x>2, and (x-6), which is positive for x>6.  So, for x<0, all 3 are negative, and the product would also be negative.  For x=0, the derivative is 0, so neither positive or negative.  For x from 0 to 2, both (x-2) and (x-6) are negative, but 3x is positive, so the answer there is also positive (negative*negative*positive = positive).  For x between 2 and 6, however, only one of the functions (x-6) is negative, so the result there is negative. Finally, for x>6 all are positive, so the derivative is also.